Integrand size = 15, antiderivative size = 13 \[ \int \frac {1+2 x^{2022}}{x+x^{2023}} \, dx=\log (x)+\frac {\log \left (1+x^{2022}\right )}{2022} \] Output:
ln(x)+1/2022*ln(x^2022+1)
Time = 1.49 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x^{2022}}{x+x^{2023}} \, dx=\log (x)+\frac {\log \left (1+x^{2022}\right )}{2022} \] Input:
Integrate[(1 + 2*x^2022)/(x + x^2023),x]
Output:
Log[x] + Log[1 + x^2022]/2022
Time = 2.55 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2026, 948, 86, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {2 x^{2022}+1}{x^{2023}+x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2 x^{2022}+1}{x \left (x^{2022}+1\right )}dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {\int \frac {2 x^{2022}+1}{x^{2022} \left (x^{2022}+1\right )}dx^{2022}}{2022}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\int \left (\frac {1}{x^{2022}}+\frac {1}{x^{2022}+1}\right )dx^{2022}}{2022}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log \left (x^{2022}\right )+\log \left (x^{2022}+1\right )}{2022}\) |
Input:
Int[(1 + 2*x^2022)/(x + x^2023),x]
Output:
(Log[x^2022] + Log[1 + x^2022])/2022
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Timed out.
hanged
Input:
int((2*x^2022+1)/(x^2023+x),x)
Output:
int((2*x^2022+1)/(x^2023+x),x)
Timed out. \[ \int \frac {1+2 x^{2022}}{x+x^{2023}} \, dx=\text {Timed out} \] Input:
integrate((2*x^2022+1)/(x^2023+x),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1+2 x^{2022}}{x+x^{2023}} \, dx=\text {Timed out} \] Input:
integrate((2*x**2022+1)/(x**2023+x),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1+2 x^{2022}}{x+x^{2023}} \, dx=\frac {1}{2022} \, \log \left (x^{2022} + 1\right ) + \log \left (x\right ) \] Input:
integrate((2*x^2022+1)/(x^2023+x),x, algorithm="maxima")
Output:
1/2022*log(x^2022 + 1) + log(x)
Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {1+2 x^{2022}}{x+x^{2023}} \, dx=\frac {1}{2022} \, \log \left (x^{2022} + 1\right ) + \frac {1}{2022} \, \log \left (x^{2022}\right ) \] Input:
integrate((2*x^2022+1)/(x^2023+x),x, algorithm="giac")
Output:
1/2022*log(x^2022 + 1) + 1/2022*log(x^2022)
Timed out. \[ \int \frac {1+2 x^{2022}}{x+x^{2023}} \, dx=\int \frac {2\,x^{2022}+1}{x^{2023}+x} \,d x \] Input:
int((2*x^2022 + 1)/(x + x^2023),x)
Output:
int((2*x^2022 + 1)/(x + x^2023), x)
Time = 0.51 (sec) , antiderivative size = 1385, normalized size of antiderivative = 106.54 \[ \int \frac {1+2 x^{2022}}{x+x^{2023}} \, dx =\text {Too large to display} \] Input:
int((2*x^2022+1)/(x^2023+x),x)
Output:
(log(x**2016 - x**2010 + x**2004 - x**1998 + x**1992 - x**1986 + x**1980 - x**1974 + x**1968 - x**1962 + x**1956 - x**1950 + x**1944 - x**1938 + x** 1932 - x**1926 + x**1920 - x**1914 + x**1908 - x**1902 + x**1896 - x**1890 + x**1884 - x**1878 + x**1872 - x**1866 + x**1860 - x**1854 + x**1848 - x **1842 + x**1836 - x**1830 + x**1824 - x**1818 + x**1812 - x**1806 + x**18 00 - x**1794 + x**1788 - x**1782 + x**1776 - x**1770 + x**1764 - x**1758 + x**1752 - x**1746 + x**1740 - x**1734 + x**1728 - x**1722 + x**1716 - x** 1710 + x**1704 - x**1698 + x**1692 - x**1686 + x**1680 - x**1674 + x**1668 - x**1662 + x**1656 - x**1650 + x**1644 - x**1638 + x**1632 - x**1626 + x **1620 - x**1614 + x**1608 - x**1602 + x**1596 - x**1590 + x**1584 - x**15 78 + x**1572 - x**1566 + x**1560 - x**1554 + x**1548 - x**1542 + x**1536 - x**1530 + x**1524 - x**1518 + x**1512 - x**1506 + x**1500 - x**1494 + x** 1488 - x**1482 + x**1476 - x**1470 + x**1464 - x**1458 + x**1452 - x**1446 + x**1440 - x**1434 + x**1428 - x**1422 + x**1416 - x**1410 + x**1404 - x **1398 + x**1392 - x**1386 + x**1380 - x**1374 + x**1368 - x**1362 + x**13 56 - x**1350 + x**1344 - x**1338 + x**1332 - x**1326 + x**1320 - x**1314 + x**1308 - x**1302 + x**1296 - x**1290 + x**1284 - x**1278 + x**1272 - x** 1266 + x**1260 - x**1254 + x**1248 - x**1242 + x**1236 - x**1230 + x**1224 - x**1218 + x**1212 - x**1206 + x**1200 - x**1194 + x**1188 - x**1182 + x **1176 - x**1170 + x**1164 - x**1158 + x**1152 - x**1146 + x**1140 - x*...